In Harry Potter, Centaurs let it be known that they are superior at divination (compared to humans).

Aside from their own boasting, or a circumstancial evidence of Dumbledore allowing Firenze to teach Divination at Hogwarts (not a high plank to fill, considering the prior "teacher"), were there any canon examples showing Centaurs successfully predicting anything specific that wasn't obvious with a little bit of logic and psychology?

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I'm not sure there could be a definitive canon answer to this question.

The ways of the centaur are shrouded in mystery. They are generally speaking as mistrustful of wizards as they are of Muggles and indeed seem to make little differentiation between us. They live in herds ranging in size from ten to fifty members. They are reputed to be well-versed in magical healing, divination, archery, and astronomy.

Fantastic Beasts and Where to Find Them - page 6 - Scholastic - An A-Z of Fantastic Beasts

Reputed means supposed to be such, so it's possible there is no specific prediction to point to in canon that is by a centaur. If I find anything, I'll edit it in.

ETA: I can't find anything that would make a good example. In Philosopher's Stone, it says:

‘Good luck, Harry Potter,’ said Firenze. ‘The planets have been read wrongly before now, even by centaurs. I hope this is one of those times.’

Firenze doesn't say what he is seeing in the planets, though, so we don't know what we're looking for as a prediction. It seems obvious that it has to do with Harry and Voldemort, with Harry not favored to triumph. Then, in Order of the Phoenix, Firenze notes the following:

‘In the past decade, the indications have been that wizardkind is living through nothing more than a brief calm between two wars. Mars, bringer of battle, shines brightly above us, suggesting that the fight must soon break out again. How soon, centaurs may attempt to divine by the burning of certain herbs and leaves, by the observation of fume and flame ...’

Order of the Phoenix - page 532 - Bloomsbury - chapter 27, The Centaur and the Sneak

I don't think this qualifies as a prediction per se; by the time of Order of the Phoenix, that another wizarding war was imminent was completely obvious. If Firenze had said this in Philosopher's Stone, I might look at it differently. Finally, in Order of the Phoenix, Firenze notes, basically, that divination should be taken with a grain of salt.

Firenze told them to look for certain shapes and symbols in the pungent fumes, but he seemed perfectly unconcerned that not one of them could see any of the signs he described, telling them that humans were hardly ever good at this, that it took centaurs years and years to become competent, and finished by telling them that it was foolish to put too much faith in such things, anyway, because even centaurs sometimes read them wrongly. He was nothing like any human teacher Harry had ever had. His priority did not seem to be to teach them what he knew, but rather to impress upon them that nothing, not even centaurs’ knowledge, was foolproof.

Order of the Phoenix - page 532 - Bloomsbury - chapter 27, The Centaur and the Sneak

I think the thing closest to a prediction is that nothing is foolproof.

It seems a bit suspicious how Firenze applies as a Divination teacher at just the moment that Professor Umbridge has sacked Professor Trelawney. Dumbledore needed a new teacher because Umbridge wanted to appoint a Ministry-influenced henchman immediately. Now I know that Dumbledore sometimes manipulates people, calls in favors, and uses theatrics, but I'd like to think he wouldn't ask Firenze to teach when both of them knew that would make Firenze a fugitive from his own herd. This gives me reason to believe that Firenze has determined from divination that Dumbledore will soon be in dire need of a new Divination teacher, and has applied to teach on his own initiative.